I wrote the following offline, and it is not a response to anything since
my last comment. I will come back later and look at what has been written
since.

This discussion has helped me see more clearly that there are two
alternate methods for describing movement across a board, each equally
valid and each useful for boards on which the other isn't. One method
describes movement in terms of the geometrical relations between spaces,
and the other describes movement in terms of the mathematical relations
between coordinates. On the usual 8x8 chessboard, not to mention any 2D
board of square spaces, these two approaches converge. The two main
geometrical relations on a square board are diagonal and lateral. A
diagonal direction is one that goes through opposite corners of a space,
while a lateral direction is one that goes through opposite sides of a
space. The mathematical relations between coordinates concern how many
axes change in the movement of a piece. In Chess, a Rook's movement
changes its place on only one axis, while a Bishop's movement changes its
place on both axes. In terms of these mathematical relations, the Rook's
movement can be described as uniaxial, and the Bishop's can be described
as biaxial. More specifically, the Bishop's movement is uniformly
biaxial. The Knight's movement is also biaxial, for it too changes its
place on both axes, but it does so unevenly, moving across one axis more
than it does the other. In Chess, a Rook's movement is both lateral and
uniaxial, and the Bishop's movement is both diagonal and uniformly
biaxial. This convergence is a coincidence caused by the fit between the
geometry and the coordinate system of the chessboard.

Let's now examine the divergence of these two approaches. A 2D hexagonal
board has two axes. In Glinkski's Hexagonal Chess, the two axes are
vertical and horizontal, as in Chess, but the horizontal axis corresponds
with diagonal, rather than lateral, lines of spaces. In the generalized
approach to hexagonal coordinates used by Game Courier, both axes describe
lateral lines of spaces, but they intersect at 60 and 120 degree angles
instead of at right angles. Whichever method of coordinates you use for a
hexagonal board, the geometrical approach and the mathematical approach no
longer converge. In Glinski's Hexagonal Chess, for example, the Bishop
sometimes moves uniaxially, and the Rook has only one line of uniaxial
movement. Using the other method, the Bishop always moves biaxially,
through not always uniformly so, while the Rook has only two lines of
uniaxial movement, and its movement across the other line is biaxial. So,
for a hexagonal board, the mathematical approach breaks down, and only the
geometrical approach is useful.

For a 3D board, the mathematical approach is useful and commonly used. In
addition to uniaxial and biaxial movement, it introduces triaxial
movement, which is movement that changes the place of a piece on all three
axes of a 3D board. Although the mathematical approach is useful for 3D
boards, the geometrical method can also be used. Diagonal movement goes
through opposite corners of a cubic space; lateral movement goes through
opposite faces; and edgewise movement goes through opposite edges.
Although both approaches can be used for a 3D board, they no longer
converge. Although lateral movement remains uniaxial, diagonal movement is
no longer biaxial. Instead, it is triaxial. In Raumschach, a well-known 3D
variant, the Bishop of Chess has been replaced by two pieces, one still
called a Bishop and the other called a Unicorn. The Raumschach Bishop
moves biaxially but not diagonally; the Unicorn moves diagonally but not
biaxially. No piece in Raumschach can move both diagonally and biaxially
at the same time.

Although either approach can be used for 3D Chess, only the mathematical
approach is really useful for 4D and higher dimensional games. The
geometrical approach is useful for both 2D and 3D games, because we can
easily visualize 2D and 3D geometrical relations. But it is much more
difficult, if not impossible, to visualize 4D relations. On a 4D tesseract
board, each space would be a tesseract, but who can visualize a tesseract?
I can't. But the mathematical approach doesn't require visualization of
multi-dimensional shapes, and it is easily adapted to endlessly multiple
dimensions. So, for a 4D game, we would just add tetraxial movement, then
pentaxial for 5D, then hexaxial for 6D, etc. In trying to play such games,
we would be pushing our own limitations, but we would not be pushing any
limitations of the mathematical model for describing piece movement. It
could adequately describe movement on boards of any number of dimensions.

One practical use of the mathematical approach is for describing the
movement of pieces to a computer. Computers have no understanding of
geometry and can do geometrical calculations only by having the geometry
reduced to mathematics. In creating ZRFs for Zillions of Games, for
example, we define directions in terms of the changes in coordinates. The
computer has no understanding of the spaces as squares, cubes, or
hexagons. All it knows are coordinates and how directions of movement
change coordinates. Despite our inability to describe the movement of
pieces to a computer using the geometrical method, it remains a perfectly
valid method for describing movement, and it is well-suited for human
understanding of piece movement.

Let me now turn to the word triagonal, which started this line of thought.
This word creates confusion, because it tries to conflate two different
methods of describing piece movement, the geometrical and mathematical.
Since it is used to describe the triaxial movement of the Unicorn, and
given that it shares 'agonal' with diagonal and 'di' is sometimes a
root meaning two while 'tri' means three, it misleadingly suggests a
contrast with diagonal. In actuality, there is no contrast between the
meaning of triagonal and diagonal. Triagonal has one of two meanings. It
either describes movement that is both triaxial and diagonal, or it
describes any triaxial movement. We can safely assume that it is not
synonomous with diagonal; otherwise, there would have been no use for this
neologism. If it describes movement that is both triaxial and diagonal, it
no more contrasts with diagonal than integer contrasts with real, for all
integers are real numbers. If triagonal is synonymous with triaxial, then
there is no more contrast between triagonal and diagonal than there is
between even and prime. A number can be both even and prime, as 2 is, or
neither, as 4 is, or odd and prime, as 1 is, or odd and not prime, as 9
is. Likewise, a line of movement can both triaxial and diagonal, neither,
or one and not the other. 

No matter which definition of triagonal we go with, it sows confusion. In
both cases, it suggests a contrast that does not exist. While the meaning
of diagonal can be found in its roots, which are 'dia' for through and
'gonal' for angles, the meaning of triagonal cannot be found in its
roots. Going by the roots of the word, all it should mean is triangular,
which does not describe a kind of movement. If triagonal describes
movement that is both triaxial and diagonal, it's a term that does not
clearly belong to either method of describing piece movement, and its use
is limited to games where triaxial movement and diagonal movement
converge. If it is synonomous with triaxial, we would avoid confusion by
abandoning the word in favor of the more accurate triaxial, whose meaning
actually is contained in its roots. Triaxial has the added advantage of
fitting into a group of terms that progressively describe movement along
increasing numbers of axes. If we followed the model of triagonal, we
might say that a 6D game has hexagonal movement, and that would be
terribly confusing. In short, using the word triagonal invites confusion,
and Gilman's description of the hexagonal Bishop's movement as triagonal
but not diagonal is evidence of this confusion. By clearly distinguishing
between the two alternate methods for describing piece movement, we can
avoid further confusion.